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http://dx.doi.org/10.4134/JKMS.j190569

TRACE EXPRESSION OF r-TH ROOT OVER FINITE FIELD  

Cho, Gook Hwa (Institute of Mathematical Sciences Ewha Womans University)
Koo, Namhun (Institute of Mathematical Sciences Ewha Womans University)
Kwon, Soonhak (Department of Mathematics Sungkyunkwan University)
Publication Information
Journal of the Korean Mathematical Society / v.57, no.4, 2020 , pp. 1019-1030 More about this Journal
Abstract
Efficient computation of r-th root in 𝔽q has many applications in computational number theory and many other related areas. We present a new r-th root formula which generalizes Müller's result on square root, and which provides a possible improvement of the Cipolla-Lehmer type algorithms for general case. More precisely, for given r-th power c ∈ 𝔽q, we show that there exists α ∈ 𝔽qr such that $$Tr{\left(\begin{array}{cccc}{{\alpha}^{{\frac{({\sum}_{i=0}^{r-1}\;q^i)-r}{r^2}}}\atop{\text{ }}}\end{array}\right)}^r=c,$$ where $Tr({\alpha})={\alpha}+{\alpha}^q+{\alpha}^{q^2}+{\cdots}+{\alpha}^{q^{r-1}}$ and α is a root of certain irreducible polynomial of degree r over 𝔽q.
Keywords
Finite field; trace; r-th root; linear recurrence relation; Tonelli-Shanks algorithm; Adleman-Manders-Miller algorithm; Cipolla-Lehmer algorithm;
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